Odd-Parity Magnets Overview

• Odd-parity magnets are defined by antisymmetric spin polarization in momentum space, distinguishing them from conventional ferromagnetic and antiferromagnetic orders.
• Recent classifications divide OPMs into collinear, coplanar, and noncoplanar types, with p-, f-, and h-wave spin splitting identified through advanced symmetry and spin-space analyses.
• OPMs exhibit unique phenomena such as electrically tunable Edelstein effects, odd-parity magnon excitations, and potential for topological superconductivity, paving the way for innovative research.

Odd-parity magnets (OPMs) are compensated magnetic states whose momentum-space spin polarization is odd under inversion, $\mathbf{S}(-\mathbf{k})=-\mathbf{S}(\mathbf{k})$, or equivalently whose nonrelativistic spin splitting satisfies $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$ in the absence of spin–orbit coupling in the paradigmatic formulations (Neumann et al., 5 Mar 2026, Luo et al., 8 Mar 2026). They are complementary to altermagnets, for which the spin splitting is even in momentum, and they support zero net magnetization despite exchange-driven band splitting (Zhang et al., 27 May 2026). The recent literature treats OPMs not as a single microscopic mechanism but as a symmetry class realized in electronic bands, magnon bands, and even orbital textures, with $p$-, $f$-, and $h$-wave form factors and with collinear, coplanar, or noncoplanar spin textures (Yu et al., 3 Jan 2025, Li et al., 20 Apr 2026).

1. Definition and conceptual scope

In the contemporary nonrelativistic usage, OPMs differ simultaneously from ferromagnets, collinear antiferromagnets, and altermagnets. Conventional ferromagnets and collinear antiferromagnets are constrained by $T$ or $PT$, while altermagnets exhibit even-parity splittings $s_n(\mathbf{k})=+s_n(-\mathbf{k})$. OPMs instead satisfy

$s_n(\mathbf{k})=-s_n(-\mathbf{k}),$

so their spin texture is antisymmetric in momentum space (Neumann et al., 5 Mar 2026). In helimagnets this appears as odd-parity spin-momentum locking of the electronic spin expectation $S_n(\mathbf{k})$, while in insulating spin models it appears as odd-wave magnon spin textures (Larsen et al., 9 Apr 2026, Neumann et al., 5 Mar 2026).

A central conceptual point is that the literature contains two complementary symmetry narratives. One widely used formulation emphasizes preserved generalized or nonsymmorphic time-reversal operations, such as the spin-translation group

$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$0

which protects $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$1 in a noncollinear compensated magnet (Neumann et al., 5 Mar 2026). A more recent reformulation argues that previous studies focused exclusively on non-relativistic spin splitting while overlooking the intrinsically broken $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$2 inherent to magnetic order, and proposes that OPMs universally host a hidden Zeeman field rooted in this $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$3-breaking (Luo et al., 16 Mar 2026). The coexistence of these descriptions is an active feature of the field rather than a settled terminological closure.

2. Symmetry criteria and classification

The unifying language of recent classification work is the spin-space group. In that notation an operation is written

$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$4

where $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$5 or $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$6, $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$7, and $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$8 is the spatial part; under $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$9, the Bloch spin texture transforms as

$p$0

with $p$1 for $p$2 and $p$3 for $p$4 (Luo et al., 8 Mar 2026). This framework classifies OPMs into three types: type-I collinear, type-II coplanar, and type-III noncoplanar. Representative symmetry elements include $p$5, which forces a collinear texture, $p$6, which forces a coplanar texture, and $p$7, which enforces odd parity under $p$8 (Luo et al., 8 Mar 2026).

The same work identifies eight distinct symmetry-driven mechanisms for OPMs and reports 33 OPM candidates from the Magndata database (Luo et al., 8 Mar 2026). A related spin-symmetry analysis likewise identifies eight symmetry-driven cases, organizes OPMs into the same three texture classes, further delineates additional symmetry requirements for $p$9-wave and $f$0-wave spin splitting in type-I systems, and reports 48 candidate materials in Magndata together with an intrinsic $f$1 topology in some OPMs (Luo et al., 7 Oct 2025). A separate antiferromagnetic-exchange program arrives at a different but complementary enumeration: 421 distinct 2D magnetic irreducible representations permitting odd-parity order, 119 minimal microscopic models, and 67 materials for which that theory applies; among those irreducible representations, spin-vector cases appear in 325 $f$2-wave, 84 $f$3-wave, and 12 $f$4-wave examples (Yu et al., 3 Jan 2025). This suggests that candidate counts are framework-dependent, but the repeated emergence of $f$5-, $f$6-, and $f$7-wave OPMs is robust across independent symmetry constructions.

3. Microscopic routes to odd-parity magnetism

Several distinct microscopic routes now exist. In the antiferromagnetic-exchange framework, the key secondary order parameter is

$f$8

which is time-reversal even and inversion odd. The same Landau theory also yields two competing translation-invariant orders, $f$9 and $h$0, leading respectively to odd-parity spin splitting, nematic order, or scalar odd-parity order related to multiferroicity (Yu et al., 3 Jan 2025). In that setting the effective spin-splitting Hamiltonian is $h$1 with $h$2.

Single-$h$3 helimagnets provide another analytically controlled route. For a magnetization satisfying

$h$4

the Generalized Bloch theorem allows the problem to be solved in the primitive cell through a twisted Hamiltonian $h$5, and magnetic-supercell quantities are recovered by reciprocal-space downfolding (Larsen et al., 9 Apr 2026). In the non-relativistic limit, coplanar spiral order always yields odd-parity magnetism: the spin component along the spiral-plane normal obeys $h$6, while $h$7. First-principles case studies on MnI$h$8, NiI$h$9, and MnTe$T$0 further show that the magnitude of the spin splitting is maximized for states having large odd-orbital ($T$1-type) character (Larsen et al., 9 Apr 2026).

More engineered routes broaden the design space. In sAFM/metal/sAFM van der Waals heterostructures, the leading RKKY-type exchange interaction is canceled by the stacking symmetry, exposing a higher-order biquadratic interaction that drives a filling-controlled transition from a collinear phase to an orthogonal $T$2-wave configuration (Kim et al., 11 Feb 2026). Floquet engineering provides a dynamic route in conventional collinear antiferromagnets: off-resonant circularly polarized light, elliptically polarized light, and bicircular light can induce odd-parity spin splitting, with CPL favoring an $T$3-wave pattern and EPL or BCL reducing it to a $T$4-wave structure (Huang et al., 28 Jul 2025). Still more unconventional realizations have been proposed in momentum-space Hatsugai–Kohmoto-like models, where OPMs appear as $T$5-wave spin-bond order with $T$6 and vanishing on-site moment (Rickelt et al., 20 Apr 2026), and in loop-current systems, where a spinless lattice model realizes $T$7-wave orbital magnetism protected by combined translation and time reversal (Li et al., 20 Apr 2026).

4. Material platforms and experimental fingerprints

Experimentally, the most direct electronic evidence so far comes from CeNiAsO, NiI$T$8, and coplanar Fe-based superconductors, while MnI$T$9, NiI$PT$0, and MnTe$PT$1 serve as first-principles helimagnetic benchmarks (Zhang et al., 27 May 2026, Song et al., 29 Apr 2025, Dsouza et al., 29 Aug 2025, Larsen et al., 9 Apr 2026).

Platform	Odd-parity fingerprint	Reported scale
CeNiAsO	one nodal plane $PT$2; minimal $PT$3	spin splitting $PT$4
NiI$PT$5	chirality-linked odd $PT$6 along $PT$7	$PT$8
Fe-based superconductors	$PT$9-wave $s_n(\mathbf{k})=+s_n(-\mathbf{k})$0 and finite nonlinear Hall response	FeSe DFT splitting up to $s_n(\mathbf{k})=+s_n(-\mathbf{k})$1

CeNiAsO is currently the clearest prototype $s_n(\mathbf{k})=+s_n(-\mathbf{k})$2-wave OPM. Below $s_n(\mathbf{k})=+s_n(-\mathbf{k})$3, the Ce moments form a coplanar, noncollinear $s_n(\mathbf{k})=+s_n(-\mathbf{k})$4 antiferromagnetic order in the $s_n(\mathbf{k})=+s_n(-\mathbf{k})$5 plane. Symmetry permits the minimal low-energy form $s_n(\mathbf{k})=+s_n(-\mathbf{k})$6, implying $s_n(\mathbf{k})=+s_n(-\mathbf{k})$7 and $s_n(\mathbf{k})=+s_n(-\mathbf{k})$8 on the nodal plane $s_n(\mathbf{k})=+s_n(-\mathbf{k})$9. Spin-resolved ARPES resolves $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$0 and $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$1 splittings of $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$2, with $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$3 reversing sign across the nodal plane; below $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$4, the magnetoresistance is two-fold, the bulk $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$5 reaches $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$6 at $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$7, and FIB devices reach $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$8 at $s_n(\mathbf{k})=-s_n(-\mathbf{k}),$9, enabling field-induced switching between high-resistance and low-resistance states through domain selection (Zhang et al., 27 May 2026).

NiI$S_n(\mathbf{k})$0 realizes the helimagnetic and multiferroic route. Noncollinear DFT on a NiI$S_n(\mathbf{k})$1 monolayer model of the bulk helix finds large spin splitting $S_n(\mathbf{k})$2 along $S_n(\mathbf{k})$3 even with SOC turned off, while $S_n(\mathbf{k})$4 vanishes along $S_n(\mathbf{k})$5 and is antisymmetric along the propagation axis (Song et al., 29 Apr 2025). Zero-bias photocurrent and circular photogalvanic measurements then directly connect the sign of the odd-parity spin texture to spiral chirality and ferroelectric polarization: below $S_n(\mathbf{k})$6, the CPGE coefficient is $S_n(\mathbf{k})$7 for one chirality and $S_n(\mathbf{k})$8 for the opposite chirality along $S_n(\mathbf{k})$9, while along the ferroelectric axis it is negligible, $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$00 (Song et al., 29 Apr 2025). The same improper-ferroelectric coupling permits voltage-based switching of the chirality and hence of the odd-parity spin polarization.

In Fe-based superconductors with coplanar magnetic order, low-energy modeling and DFT identify an $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$01-wave OPM. The leading spin-splitting form factor is

$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$02

or, in lattice form for FeSe,

$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$03

DFT fits indicate splittings of order a few meV in LaFeAsO for realistic $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$04, while in FeSe the splitting can reach $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$05 (Dsouza et al., 29 Aug 2025, Yu et al., 3 Jan 2025). A related and experimentally important point is that these OPMs support finite out-of-plane Berry curvature and a nonlinear anomalous Hall effect even though the linear Hall effect vanishes (Dsouza et al., 29 Aug 2025).

5. Magnons and odd-parity bosonic excitations

Odd-parity magnetism is not restricted to electronic quasiparticles. In insulating noncoplanar compensated magnets, minimal spin Hamiltonians built only from isotropic Heisenberg bilinear and biquadratic terms produce magnon bands with odd-wave spin textures and no relativistic interactions (Neumann et al., 5 Mar 2026). Three minimal examples are particularly important: a vector $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$06-wave OPM on the kagome lattice, a $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$07-wave antialtermagnet with collinear $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$08, and an $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$09-wave antialtermagnet on stacked triangular layers with three nodal planes and collinear magnon spin along the $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$10 axis. These models show that antialtermagnetism can arise in the magnon sector even when the ground state is noncoplanar (Neumann et al., 5 Mar 2026).

The magnon spin is obtained from the paraunitary Bogoliubov transformation of the linear spin-wave Hamiltonian, and its odd-wave character immediately produces nonequilibrium responses. In particular, the thermal Edelstein effect is

$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$11

with $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$12 determined by the magnon spin polarization, magnon velocity, magnon energy, and the derivative of the Bose function (Neumann et al., 5 Mar 2026). In the $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$13-wave antialtermagnet only $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$14, giving a two-lobed $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$15-orbital angular pattern; the vector $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$16-wave model instead yields an in-plane response, while the $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$17-wave case would produce a six-fold anisotropy (Neumann et al., 5 Mar 2026).

A second bosonic route appears in the Haldane–Hubbard model. There, topological spin-flip excitons in the paramagnetic phase condense into a collinear Néel state, and the resulting magnon bands acquire odd-parity $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$18-wave splitting (Eto et al., 4 Jun 2026). In the topological antiferromagnetic phase the two magnon bands carry Chern number $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$19; an electronic band-gap closing enforces a magnon gap closing and changes the odd-parity magnon topology (Eto et al., 4 Jun 2026). This establishes odd-parity magnons as both a symmetry phenomenon and a topological one.

6. Response functions, topology, and active directions

The signature response of electronic OPMs is the non-relativistic Edelstein effect. In a coplanar single-$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$20 helimagnet, an electric field shifts the Fermi surface, $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$21, and the antisymmetry $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$22 produces a net nonequilibrium spin, summarized in the chain $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$23 (Larsen et al., 9 Apr 2026). The same Generalized Bloch theorem framework extends directly to Kubo-type linear and nonlinear response functions without ever constructing large magnetic supercells (Larsen et al., 9 Apr 2026). In van der Waals heterostructures, the orthogonal $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$24-wave phase exhibits a gate-tunable Edelstein tensor $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$25, and the response remains qualitatively unchanged even for Rashba SOC as large as $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$26 (Kim et al., 11 Feb 2026).

At the same time, OPMs do not exhibit a universal Edelstein response. In the Fe-based $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$27-wave OPMs, the Edelstein effect vanishes without SOC because $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$28 exactly and mirror symmetries force $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$29; only after adding atomic SOC do in-plane $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$30-wave spin components and a finite in-plane Edelstein response appear (Dsouza et al., 29 Aug 2025). This is an important correction to the common expectation that odd parity in momentum space automatically implies current-induced spin polarization.

Topological and adjacent extensions are now central. One spin-symmetry analysis shows that OPMs can exhibit an intrinsic $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$31 topology (Luo et al., 7 Oct 2025). A different proposal argues that OPMs are an ideal platform for topological superconductivity because the large non-relativistic spin splitting coexists with a hidden Zeeman field and can support chiral or unidirectional Majorana boundary modes in the presence of $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$32-wave pairing (Luo et al., 16 Mar 2026). Beyond spin, loop-current models demonstrate $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$33-wave orbital magnetism with zero macroscopic orbital magnetization but finite orbital Hall conductivity (Li et al., 20 Apr 2026). Quantum-geometric mechanisms add yet another layer: on the bilayer Lieb lattice, the quantum metric enhances ferroic odd-parity multipole fluctuations, and an on-site Hubbard interaction drives their RPA condensation into long-range odd-parity multipole order (Kudo et al., 27 May 2025).

Three misconceptions are now difficult to sustain. First, OPMs are not limited to noncollinear magnets: they can occur in collinear, coplanar, and noncoplanar orders, and can even be induced dynamically in conventional collinear antiferromagnets (Luo et al., 8 Mar 2026, Huang et al., 28 Jul 2025). Second, OPMs are not restricted to $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$34-wave spin textures: $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$35-wave and $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$36-wave forms are explicit in both symmetry classifications and material realizations (Yu et al., 3 Jan 2025, Dsouza et al., 29 Aug 2025). Third, OPMs are not exclusively electronic-spin phenomena: magnons, orbital moments, and odd-parity multipole fluctuations all realize closely related symmetry structures (Neumann et al., 5 Mar 2026, Li et al., 20 Apr 2026, Kudo et al., 27 May 2025).

The field has therefore moved from a narrow definition of exchange-driven odd spin splitting to a broader program of symmetry-guided design. Current directions emphasize primitive-cell treatments of helimagnets, electrically controlled heterostructures, multiferroic switching, odd-parity magnons, and field-free topological superconductivity, while the precise relation between generalized time-reversal descriptions and hidden-$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$37-breaking descriptions remains a central conceptual issue (Larsen et al., 9 Apr 2026, Luo et al., 16 Mar 2026).